Three-manifolds with positive Ricci curvature
In this paper the following important theorem is proved: Let \(X\) be a compact 3-manifold which admits a Riemannian metric with strictly positive Ricci curvature. Then \(X\) also admits a metric of constant positive curvature. As all manifolds of constant curvature have been completely classified by \textit{J. A. Wolf} in his book [Spaces of constant curvature. New York etc.: McGraw-Hill (1967; Zbl 0162.53304)], by the preceding theorem these are the only compact three-manifolds, which can carry metrics of strictly positive Ricci curvature. Accordingly, a conjecture of \textit{J.-P. Bourguignon} [Global differential geometry and global analysis, Proc. Colloq., Berlin 1979, Lect. Notes Math. 838, 249--250 (1981; Zbl 0437.53011)] is answered affirmatively. As pointed out by the author, this theorem is closely related to the famous Poincaré's conjecture on the compact, simply-connected 3-manifolds and the Smith's conjecture related to the group of covering transformations. If both conjectures were true, the main theorem mentioned above would follow as a corollary. In the proof of this theorem the author studied in advance the equation of evolution \(\frac{\partial g_{ij}}{\partial t}=\frac 2n rg_{ij} - 2R_{ij}\), where \(r\) is the average of the scalar curvature \(R\), namely \(r=\int R\,d\mu/\int\,d\mu\). He proved that if for a compact 3-manifold the initial metric has strictly positive Ricci curvature, then it continues like that for ever, and converges to a metric of constant positive curvature, as \(t\) tends to \(\infty\). Then he made use of the fact peculiar to three dimensions, that the full Riemannian curvature tensor \(R_{hijk}\) can be calculated from the Ricci tensor \(R_{ij}\) (Weyl's conformal curvature tensor is vanishing). In view of this crucial step, the method used here cannot be generalized to the case \(\dim(M)\geq 3\), unless it is extensively modified.
scientific article; zbMATH DE number 3794902
scientific article